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Indefinite Integrals of Polynomials

We are now going to look at a technique for finding the indefinite integrals of the simplest type of functions - polynomials!

Theorem 1: If $f$ is in the form $f(x) = bx^n$ where $b$ and $C$ are both constants, then $\int bx^n \: dx = \frac{bx^{n+1}}{n + 1} + C$.

Proof: From the fundamental theorem of calculus part 1, we get that $\frac{d}{dx} \int f(x) \: dx = f(x)$. If we can differentiate both sides of $\int bx^n \: dx = \frac{bx^{n+1}}{n + 1} + C$ and prove their equality, then we have proven the equality of the integral.